Thursday, February 6, 2014

Eratosthenes' Earth, 2200 BC

Eratosthenes: What curious minds do best

More than 2,200 years ago the Greek mathematician
Eratosthenes was able to calculate the circumference of the Earth by using
geometry, deductive reasoning and some carefully considered assumptions among
astronomers of the Hellenistic period. It
was the Greek astronomer Aristarchus that presented the earliest known model of
the Earth orbiting around the Sun and his ideas would have been known to
Eratosthenes and he would build upon this foundation.

If the Sun was at the center then there was a philosophical
inclination to believe it was also likely to be vastly larger than its orbiting
bodies such as the Earth, or its Moon.
The Greeks had other, more objective, reasons for this belief and they
used this and other evidence in an attempt to determine the distance between
the Sun and our Earth. The numbers they
arrived at were never close to the actual truth as their means were too
rudimentary and the idea of a burning sphere warming a planet from ninety-three
million miles away was likely to seem absurd.
Eratosthenes believed that whatever the actual distance turned out to
be, it would be great enough for the Sun’s rays to strike the Earth parallel to
one another. This would be a crucial
factor in Eratosthenes’ geometric-based calculation of the Earth’s
circumference.

Numbers off but right approach

Another belief important to Eratosthenes and other
astronomers was the conclusion that the Earth was a sphere. The evidence of this was revealed by the
curve of the Earth’s shadow that was cast upon the Moon’s face during a lunar
eclipse. There were more subtle forms of
evidence supporting this theory, such as a departing ship’s image descending
beneath the horizon, but certainly the shape of the Earth’s form projected upon
the Moon was the most dramatic.

Finding the arc's size in relation to the circle

As it happened Eratosthenes was in Egypt and he had become
aware that on the first day of summer, at high noon, the Sun was directly
overhead in a place named Syene, now Aswan, along the Nile. The Sun’s rays were perpendicular to the land
and they shone straight down a deep well, reflecting off the water at its normally
darkened depth. Eratosthenes was also
able to determine that at the same time on the same day the Sun’s rays were at an
angle of just over seven degrees from the perpendicular in Alexandria, another
location in Egypt. He calculated this by
measuring the shadow cast at noon in this Mediterranean town. Knowing that a seven plus degree angle
represented 1/50th of the circumference of a circle Eratosthenes now
needed to know the distance from Syene to Alexandria. His calculation would make that length about
925 kilometers in modern measurement terms.
The actual length is closer to 800 kilometers, introducing one of a
number of inaccuracies. A related
inaccuracy had to do with the precision of the determined angle in
Alexandria. The length of the arc
representing the distance between the two Egyptian locations should have represented
1/48th the circumference of the Earth and not 1/50th.

Fraction of circle times arc's length gives circumference

The circumference of the Earth, according to Eratosthenes,
would be 46,250 kilometers or 28,721 miles using contemporary measurement
standards. It is thought Eratosthenes
used the Greek measurement of stadion, believed to approximate 185 meters. This
adds to the uncertainty of Eratosthenes’ final value for the Earth’s size. The Earth’s actual circumference around the
equator is a bit over forty thousand kilometers or 24,902 miles. Had Eratosthenes’ calculation given 800
kilometers as the distance between Syene and Alexandria his numbers would come
astonishingly close to our current measurement, with only 62 miles separating
the two figures.

A kilometer is just over half a mile

In any case, the human mind demonstrated extraordinary
reasoning ability twenty-two hundred years ago, armed only with primitive tools
and the most limited of inherited knowledge.
Here were individuals attempting to understand their world, not through
metaphors and stories, but through objective observation of the world about
them and using the tool of mathematics which would improve with time in both
power and sophistication.